What does this statement mean? Pertaining to Definition of Lipschitz continuous function

What does this statement mean? Pertaining to Definition of Lipschitz
continuous function

I'm being introduced into the Picard Existence Theorem. I am fairly
comfortable with math terminology but not great at it. In your answer I
would appreciate a mathematicians answer, sparsed with some actual english
please, to help explain things =).
The statement is defining a Lipschitz continuous function. Here is the
statement.
A function $f: U \times [t_0;t_0 + T ] \rightarrow \mathbb{R}^n, U \subset
\mathbb{R}^n$, is Lipschitz continuous in $U$ if there exists a constant
$L$ such that $\| f(y,t) - f(x,t) \| \le L\|x - y\|$ for all $x,y \in U$
and $t \in [t_0;t_0 + T ]$. If $U = \mathbb{R}^n$, $f$ is called globally
Lipschitz.
What I don't get:
1) The $\times$ after the first $U$. Is that saying the cartesian product
of $U$ and $[t_0;t_0 + T ]$? I get the underlying meaning of this
statement...I have a function that is mapping $U$ into $\mathbb{R}^n$ but
I don't get the specifics of that first part.
2) The second statement I understand the math this is the part I'm most
confused about, but don't understand the meaning of it. I mean what is
this constant $L$ in the first place? I just don't see where this
statement comes from or what it even means.
The rest of the statement I understand, thanks for the help!